Cook rice: My preferred method is to add the rice, water, and oil to the Instant Pot and hit the rice button (high pressure 12 minutes, quick release pressure). Perfect results every time. You can also use a rice cooker or stovetop.
Chill rice: Line a large baking sheet with parchment paper. Transfer hot rice to the lined baking sheet, press into a single layer, and chill in the refrigerator 15-60 minutes.
Prepare ingredients: Because this is a quick-cooking recipe, I like to have all of the ingredients chopped, partitioned, and ready to go. Using several bowls, combine diced onion and carrot in one bowl, chopped broccoli in another, and the beaten eggs in another. Cut the boneless skinless chicken thighs into very small pieces. This is easier if the meat is partially frozen and you use a very sharp knife. Place the chicken in a separate bowl.
Cook onion and carrot: In a very large pan or wok (preferred) over medium-high to high heat, melt 1 Tablespoon of the butter. Add the diced onion and carrot. Toss to combine in butter and then only stir to prevent burning. You want them to get slightly seared. Once they get slightly soft and have some nice color, transfer them back to their bowl.
Cook broccoli: Add another tablespoon of butter to the pan. Sauté the broccoli stirring only occasionally. When vibrant green and slightly browned, transfer to the bowl with the onions and carrots.
Cook chicken and garlic: Add 2 tablespoons of butter to the pan. Add the chicken pieces and minced garlic. Toss to coat in butter and then spread into a single layer to maximize contact with the bottom of the pan. To get a nice sear, stir the chicken only occasionally to promote even cooking and prevent burning. The mixture will release liquid as it cooks and won't begin to sear until all of the liquid has cooked off. When done, transfer browned garlic chicken to the bowl with the sauteed vegetables (do not return to bowl raw chicken was in).
Cook egg: Add beaten egg to the hot pan and swirl to create a thin layer. Cook just until scrambled and transfer to bowl with other cooked ingredients.
Heat rice: Add the remaining 4 tablespoons of butter to the pan. Add the cold rice and spread to an even layer. Top with a swirl of toasted sesame oil. Allow the rice to cook for a couple of minutes undisturbed before stirring to get the bottom layer to crisp up a bit. Add the soy sauce, allow it to cook for another minute or so undisturbed, then give it a good stir to heat through.
Combine ingredients: Add all of the meat, scrambled egg, and veggies back to the pan (along with any liquid from the bowl). Stir to combine to heat through. Taste and add additional toasted sesame oil and/or soy sauce, as well as salt and pepper, if desired. Green onions and sesame seeds can be mixed in or sprinkled on top as garnish.
Serve: Enjoy hot as is or with some chili paste or crushed peanuts.